Introduction to Programming and 4Algorithms Abstract Types. Uwe R. Zimmer - The Australian National University

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1 Introduction to Programming and 4Algorithms 2015 Uwe R. Zimmer - The Australian National University

2 [ Thompson2011 ] Thompson, Simon Haskell - The craft of functional programming Addison Wesley, third edition 2011 References 2015 Uwe R. Zimmer, The Australian National University page 335 of 794 (chapter 4: up to page 369)

3 Definition An abstract type is defined by 1. A type name the type structure can be partly revealed or stay completely hidden and can be as simple as a character or as complicated as a database. 2. Functions / Operations which work on this type and have to follow defined rules, like an e.g. arithmetic laws or a specific algebra Uwe R. Zimmer, The Australian National University page 336 of 794 (chapter 4: up to page 369)

4 Definition An abstract type is defined by 1. A type name the type structure can be partly revealed or stay completely hidden and can be as simple as a character or as complicated as a database. 2. Functions / Operations which work on this type and have to follow defined rules, like an e.g. arithmetic laws or a specific algebra. What s abstract? We don t necessarily know how values are stored in memory or how the operations are implemented. we can for instance use a floating point number without knowing which bit goes where, or how the sine function is implemented 2015 Uwe R. Zimmer, The Australian National University page 337 of 794 (chapter 4: up to page 369)

5 Definition An abstract type is defined by 1. A type name the type structure can be partly revealed or stay completely hidden and can be as simple as a character or as complicated as a database. 2. Functions / Operations which work on this type and have to follow defined rules, like an e.g. arithmetic laws or a specific algebra. What s abstract? We don t necessarily know how values are stored in memory or how the operations are implemented. All good programming interfaces are designed on a need to know basis. we can for instance use a floating point number without knowing which h bit goes where, or how the sine function is implemented 2015 Uwe R. Zimmer, The Australian National University page 338 of 794 (chapter 4: up to page 369)

6 Enumerations: Basic types Basic, scalar types overview Boolean (Bool), Characters (Char) Numbers: Cardinal (-), Integer (Integer), Integer range (Ix), Rational (Rational), Real (Float, Double), Complex (Complex) Machine related types: Bits (Bits), N Words: Integer range 0f 2-1 (Word, WordN), N- Two s complements: Integer range 2 1 N- - f (Int, IntN) with N! 2015 Uwe R. Zimmer, The Australian National University page 339 of 794 (chapter 4: up to page 369)

7 Enumerations Boolean (Bool) Constructors: data Bool = False True Base functions: and (&&), or ( ), not (not) (&&), ( ) :: Bool -> Bool -> Bool not :: Bool -> Bool 2015 Uwe R. Zimmer, The Australian National University page 340 of 794 (chapter 4: up to page 369)

8 Enumerations Boolean (Bool) Constructors: data Bool = False True Base functions: and (&&), or ( ), not (not) (&&), ( ) :: Bool -> Bool -> Bool not :: Bool -> Bool Some related functions: (==), (/=) :: Eq a => a -> a -> Bool (<), (<=), (>=), (>) :: Ord a => a -> a -> Bool isnan, isinfinite, isdenormalized, isnegativezero, isieee :: RealFloat a => a -> Bool 2015 Uwe R. Zimmer, The Australian National University page 341 of 794 (chapter 4: up to page 369)

9 Enumerations Boolean (Bool) Constructors: data Bool = False True Base functions: and (&&), or ( ), not (not) (&&), ( ) :: Bool -> Bool -> Bool not :: Bool -> Bool Some related functions: (==), (/=) :: Eq a => a -> a -> Bool (<), (<=), (>=), (>) :: Ord a => a -> a -> Bool isnan, isinfinite, isdenormalized, isnegativezero, isieee :: RealFloat a => a -> Bool Example functions: threshold_reached :: Integer -> Bool threshold_reached number = number > 100 and_3 :: Bool -> Bool -> Bool -> Bool and_3 x y z = x && y && z 2015 Uwe R. Zimmer, The Australian National University page 342 of 794 (chapter 4: up to page 369)

10 Enumerations Boolean (Bool) Example function Exclusive or : ex_or1 :: Bool -> Bool -> Bool ex_or1 x y = case (x, y) of (False, False) -> False (False, True ) -> True (True, False) -> True (True, True ) -> False 2015 Uwe R. Zimmer, The Australian National University page 343 of 794 (chapter 4: up to page 369)

11 Enumerations Boolean (Bool) Example function Exclusive or : ex_or1 :: Bool -> Bool -> Bool ex_or1 x y = case (x, y) of (False, False) -> False (False, True ) -> True (True, False) -> True (True, True ) -> False ex_or2 :: Bool -> Bool -> Bool ex_or2 x y = case x of False -> y True -> not y 2015 Uwe R. Zimmer, The Australian National University page 344 of 794 (chapter 4: up to page 369)

12 Enumerations Boolean (Bool) Example function Exclusive or : ex_or1 :: Bool -> Bool -> Bool ex_or1 x y = case (x, y) of (False, False) -> False (False, True ) -> True (True, False) -> True (True, True ) -> False ex_or2 :: Bool -> Bool -> Bool ex_or2 x y = case x of False -> y True -> not y ex_or3 :: Bool -> Bool -> Bool ex_or3 x y = (x && not y) (not x && y) 2015 Uwe R. Zimmer, The Australian National University page 345 of 794 (chapter 4: up to page 369)

13 Enumerations Boolean (Bool) Example function Exclusive or : ex_or1 :: Bool -> Bool -> Bool ex_or1 x y = case (x, y) of (False, False) -> False (False, True ) -> True (True, False) -> True (True, True ) -> False ex_or2 :: Bool -> Bool -> Bool ex_or2 x y = case x of False -> y True -> not y ex_or4 :: Bool -> Bool -> Bool ex_or4 x y = x /= y ex_or3 :: Bool -> Bool -> Bool ex_or3 x y = (x && not y) (not x && y) 2015 Uwe R. Zimmer, The Australian National University page 346 of 794 (chapter 4: up to page 369)

14 Enumerations Boolean (Bool) Testing example implementations: import Test.QuickCheck test_ex_ors :: Bool -> Bool -> Bool test_ex_ors x y = (ex_or1 x y) == (ex_or2 x y) && (ex_or2 x y) == (ex_or3 x y) && (ex_or3 x y) == (ex_or4 x y) > quickcheck test_ex_ors +++ OK, passed 100 tests Uwe R. Zimmer, The Australian National University page 347 of 794 (chapter 4: up to page 369)

15 Enumerations Boolean (Bool) Boolean algebra: (a && b) && c == a && (b && c) (a b) c == a (b c) (associative) a && b == b && a a b == b a (commutative) a && (a b) == a a (a && b) == a (absorption) a && (b c) == (a && b) (a && c) a (b && c) == (a b) && (a c) (distribution) a && not a == False a not a == True (complement) Those basic laws (which are one way to axiomatize the boolean algebra) may help to simplify logic expressions. Compilers might use them as well to provide warnings like This guard expression will always be False, hence the following code is unreachable Uwe R. Zimmer, The Australian National University page 348 of 794 (chapter 4: up to page 369)

16 Enumerations Boolean (Bool) Constructors: data Bool = False True Base functions: and (&&), or ( ), not (not) (&&), ( ) :: Bool -> Bool -> Bool not :: Bool -> Bool Algebra: (a && b) && c == a && (b && c) (a b) c == a (b c) (associative) a && b == b && a a b == b a (commutative) a && (a b) == a a (a && b) == a (absorption) a && (b c) == (a && b) (a && c) a (b && c) == (a b) && (a c) (distribution) a && not a == False a not a == True (complement) 2015 Uwe R. Zimmer, The Australian National University page 349 of 794 (chapter 4: up to page 369)

17 Enumerations Boolean (Bool) Constructors: data Bool = False True Base functions: and (&&), or ( ), not (not) (&&), ( ) :: Bool -> Bool -> Bool not :: Bool -> Bool An abstract type is defined by its name plus operations (functions) which have to follow certain rules (algebra). Algebra: (a && b) && c == a && (b && c) (a b) c == a (b c) (associative) a && b == b && a a b == b a (commutative) a && (a b) == a a (a && b) == a (absorption) a && (b c) == (a && b) (a && c) a (b && c) == (a b) && (a c) (distribution) a && not a == False a not a == True (complement) 2015 Uwe R. Zimmer, The Australian National University page 350 of 794 (chapter 4: up to page 369)

18 Enumerations Character (Char) Definition: data Char = Enumeration of Unicode Characters (ISO/IEC 10646) in UTF-32 first 128 characters are identical to ASCII, first 256 characters are identical to ISO (Latin-1). Base functions: iscontrol, isspace, islower, isupper, isalpha, isalphanum, isprint, isdigit, isoctdigit, ishexdigit, isletter, ismark, isnumber, ispunctuation, issymbol, is- Separator, isascii, islatin1, isasciiupper, isasciilower :: Char -> Bool generalcategory :: Char -> GeneralCategory (Emumeration of Unicode categories) toupper, tolower, totitle :: Char -> Char digittoint :: Char -> Int inttodigit :: Int -> Char (works for hexadecimal characters / values too) ord :: Char -> Int chr :: Int -> Char 2015 Uwe R. Zimmer, The Australian National University page 351 of 794 (chapter 4: up to page 369)

19 Definition: Numbers Integer numbers (Integer) Integer numbers negative, zero, positive no bounds! Base functions: max, min :: Ord a => a -> a -> a (+), (-), (*), (^) :: Num a => a -> a -> a quot, rem, div, mod :: Integral a => a -> a -> a quotrem, divmod :: Integral a => a -> a -> (a, a) negate, abs, signum :: Num a => a -> a frominteger :: Num a => Integer -> a tointeger :: Integral a => a -> Integer 2015 Uwe R. Zimmer, The Australian National University page 352 of 794 (chapter 4: up to page 369)

20 Numbers Integer numbers (Integer) Definition: Integer numbers negative, zero, positive no bounds! Base functions: max, min :: Ord a => a -> a -> a (+), (-), (*), (^) :: Num a => a -> a -> a quot, rem, div, mod :: Integral a => a -> a -> a quotrem, divmod :: Integral a => a -> a -> (a, a) negate, abs, signum :: Num a => a -> a frominteger :: Num a => Integer -> a tointeger :: Integral a => a -> Integer a stands for multiple types here including Integer quot, rem truncate towards zero div, mod truncate towards negative infinity 2015 Uwe R. Zimmer, The Australian National University page 353 of 794 (chapter 4: up to page 369)

21 Numbers Ranges and Modulo Types (Ix) Definition: Sub-ranges of discrete types (Integer or enumerations), either with an out of range detection or a modulo arithmetic semantic. Not directly available in Haskell. (yet e.g. the Ix class implements part of the functionality.) Base functions: range :: Ord a => (a, a) -> [a] index :: Ord a => (a, a) -> a -> Int inrange :: Ord a => (a, a) -> a -> Bool rangesize :: Ord a => (a, a) -> Int a stands for any type on which an order is defined including Integer Example expressions: > range ( a, z ) abcdefghijklmnopqrstuvwxyz > inrange ( a, z ) b True > index ( a, z ) b 1 > range (1, 12) [1,2,3,4,5,6,7,8,9,10,11,12] Indices start at Uwe R. Zimmer, The Australian National University page 354 of 794 (chapter 4: up to page 369)

22 Numbers Ranges and Modulo Types (Ix) Definition: Sub-ranges of discrete types (Integer or enumerations), either with an out of range detection or a modulo arithmetic semantic. Not directly available in Haskell. (yet e.g. the Ix class implements part of the functionality.) Range and modulo type examples: Natural = subtype Integer range 0.. Positive = subtype Integer range 1.. Index = subtype Positive range Base_Colour = subtype Color range Red.. Blue Circular_Index = modulo 10 (none of which are natively available in Haskell) 2015 Uwe R. Zimmer, The Australian National University page 355 of 794 (chapter 4: up to page 369)

23 Numbers Rational numbers (Rational) Definition: data Integral a => Ratio a type Rational = Ratio Integer Any number defined by the ratio of two Integers no bounds or precision limits! Base functions: (%) :: Integer -> Integer -> Rational numerator :: Rational -> Integer denominator :: Rational -> Integer approxrational :: RealFrac a => a -> a -> Rational Real number which h needs converting Precision of conversion Example expressions: > 10 * (3 % 4) 15 % 2 > approxrational pi % 7 frominteger (numerator api) / frominteger (denominator api) where api = approxrational pi Uwe R. Zimmer, The Australian National University page 356 of 794 (chapter 4: up to page 369)

24 Numbers Real numbers (Float, Double) Definition: Limited precision real numbers in floating point notation (IEEE 754). - Float is stored in 32 bits (about 7 significant digits and approximate range:! 10 f10 - Double is stored in 64 bits (about 16 significant digits and approx. range:! 10 f Base functions (in addition to previously introduced numerical functions): pi :: Floating a => a exp, log, sqrt, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh :: Floating a => a -> a (**), logbase, atan2 :: Floating a => a -> a -> a (^^) :: (Fractional a, Integral b) => a -> b -> a isnan, isinfinite, isdenormalized, isnegativezero, isieee :: RealFloat a => a -> Bool truncate, round, ceiling, floor :: Integral b => a -> b ). ) Uwe R. Zimmer, The Australian National University page 357 of 794 (chapter 4: up to page 369)

25 Numbers Complex numbers (Complex) Definition: data RealFloat a => Complex a = a :* a Limited precision complex numbers based on the type Float or Double for its components. Base functions (in addition to previously introduced numerical functions): realpart, imagpart, magnitude, phase :: Complex a -> a conjugate :: Complex a -> Complex a polar :: Complex a -> (a, a) mkpolar :: a -> a -> Complex a cis :: a -> Complex a 2015 Uwe R. Zimmer, The Australian National University page 358 of 794 (chapter 4: up to page 369)

26 Machine oriented scalars Words as Cardinals/Naturals (WordN) Definition: Word, Word8, Word16, Word32, Word64 are representations for words in memory interpreted as natural numbers between 0 and 2 N - 1 for WordN. Word stands for a word of default size for the local machine (size can be checked at runtime). Base functions: Functions are identical to the functions for the Integers. Warning note: Words have limited precision and all arithmetic is performed in mod 2 N. -4 :: Word (65500 :: Word16) + (40 :: Word16) 4 (65500 :: Integer) + (40 :: Integer) (65540 :: Integer) `mod` (2 ^ 16) 4 No error or warning is raised in any of those expressions in Haskell, C or Java! 2015 Uwe R. Zimmer, The Australian National University page 359 of 794 (chapter 4: up to page 369)

27 Machine oriented scalars Words as Integers (IntN) Definition: Int, Int8, Int16, Int32, Int64 are representations for words in memory interpreted as integer numbers between -2 N -1 N - and for IntN (2 s complement representation). Int has the same size as Word (size can be checked at runtime). Base functions: Functions are identical with functions for Integer. Warning note: Integers have limited precision and all arithmetic is performed in warp-around mode: (127 :: Int8) (127 :: Int8) (127 :: Integer) (((127 :: Integer) ^7) `mod` 2^8) - 2^7-128 No error or warning is raised in any of those expressions in Haskell, C or Java! 2015 Uwe R. Zimmer, The Australian National University page 360 of 794 (chapter 4: up to page 369)

28 Machine oriented scalars Bits (Bits) Definition: The type Bits is commonly used to manipulate Word or Int types on bit level. Base functions: (.&.), (..), xor :: a -> a -> a complement :: a -> a bit :: Int -> a shift, rotate, shiftl, shiftr, rotatel, rotater, setbit, clearbit, complementbit :: a -> Int -> a testbit :: a -> Int -> Bool issigned :: a -> Bool bitsize :: a -> Int Typical usages: Interface programming Efficient implementation of flag-arrays or bit-vectors Example: shift (8 :: Word8) Uwe R. Zimmer, The Australian National University page 361 of 794 (chapter 4: up to page 369)

29 Machine oriented scalars Bits (Bits) Definition: The type Bits is commonly used to manipulate Word or Int types on bit level. Base functions: (.&.), (..), xor :: a -> a -> a complement :: a -> a bit :: Int -> a shift, rotate, shiftl, shiftr, rotatel, rotater, setbit, clearbit, complementbit :: a -> Int -> a testbit :: a -> Int -> Bool issigned :: a -> Bool bitsize :: a -> Int Example: shift (8 :: Word8) 2 32 Bits type is outside the scope of this course only mentioned for completeness. Typical usages: Interface programming Efficient implementation of flag-arrays or bit-vectors 2015 Uwe R. Zimmer, The Australian National University page 362 of 794 (chapter 4: up to page 369)

30 Basic scalar types Choosing the best type In case of boolean and character types: no choice. In case of numerical types: 2015 Uwe R. Zimmer, The Australian National University page 363 of 794 (chapter 4: up to page 369)

31 Basic scalar types Choosing the best type Mathematical types: (Natural), (Positive), Integer, Rational: Can express any precision value: more_precise_pi = % No loss of precision during calculations. Behave exactly as the numbers by this name in mathematics. Versatile types: Float, Double, Complex: Limited precision, compact representation. Precision loss during calculations. Efficient calculations incl. exponential and trigonometric functions. Choice of precision. Machine oriented types: Word, Int, Bits: Most efficient calculations. Verification required that all calculations stay within limits of their types (manually or by means of automated run-time checks) Uwe R. Zimmer, The Australian National University page 364 of 794 (chapter 4: up to page 369)

32 Repetition: 1. A type name Definition An abstract type is defined by the type structure can be partly revealed or stay completely hidden and can be as simple as a character or as complicated as a database. 2. Functions / Operations which work on this type and have to follow defined rules, like an e.g. arithmetic laws or a specific algebra. What s abstract? We don t necessarily know how values are stored in memory or how the operations are implemented. All good programming interfaces are designed on a need to know basis. we can for instance use a floating point number without knowing which h bit goes where, or how the sine function is implemented 2015 Uwe R. Zimmer, The Australian National University page 365 of 794 (chapter 4: up to page 369)

33 Enumerations: Basic types Basic, scalar types overview Boolean (Bool), Characters (Char) Numbers: Cardinal (-), Integer (Integer), Integer range (Ix), Rational (Rational), Real (Float, Double), Complex (Complex) Machine related types: Bits (Bits), N Words: Integer range 0f 2-1 (Word, WordN), N- Two s complements: Integer range 2 1 N- - f (Int, IntN) with N! 2015 Uwe R. Zimmer, The Australian National University page 366 of 794 (chapter 4: up to page 369)

34 Enumerations: Basic types Basic, scalar types overview Boolean (Bool), Characters (Char) Numbers: Cardinal (-), Integer (Integer), Integer range (Ix), Rational (Rational), Real (Float, Double), Complex (Complex) Machine related types: Bits (Bits), N Words: Integer range 0f 2-1 (Word, WordN), N- Two s complements: Integer range 2 1 N- - f (Int, IntN) with N! Can you envision other abstract types? 2015 Uwe R. Zimmer, The Australian National University page 367 of 794 (chapter 4: up to page 369)

35 Enumerations: Basic types Basic, scalar types overview Boolean (Bool), Characters (Char) Numbers: Cardinal (-), Integer (Integer), Integer range (Ix), Rational (Rational), Real (Float, Double), Complex (Complex) Machine related types: Bits (Bits), Words: Integer range 0f 2 N - 1 (Word, WordN), N- Two s complements: Integer range f 2 N -1 (Int, IntN) with N! 6 Can you envision other abstract types? There will be many in fact every program can be expressed as a set of abstract types we will look into some more complex types next Uwe R. Zimmer, The Australian National University page 368 of 794 (chapter 4: up to page 369)

36 Definition What is abstract in an abstract type? Examples of abstract type Summary Boolean (Bool), Characters (Char). Numbers: Cardinal (-), Integer (Integer), Integer range (Ix), Rational (Rational), Real (Float, Double), Complex (Complex). Machine related types: Bits (Bits), Words: Integer range (Word, WordN), Two s complements Uwe R. Zimmer, The Australian National University page 369 of 794 (chapter 4: up to page 369)